Nicolis G and Nicolis C, indicate in the book Foundations of Complex Systems (2012)[1]; that the question whether deterministic dynamical systems may generate behaviors similar to the Hurst phenomenon remains at this stage completely open. However, it seems important to mention that some preliminaries calculations of the Hurst exponent have been done for a Lorenz deterministic dynamical system[2] (Mata, 1991[3], 1993[4]); particularly for the study of a time series of the intensity of convective motion, in order to visualize the potential existence of persistence with (0.64 < Hurts exponent <0.78) or low-frequency variability for this Lorenz system’s[5].

Figure 1: Rydzyna Castle. Originally, it was built at the beginning of 15th century. It was one of the most splendid palaces in Great Poland (1704-1709) residences of Polish king Stanislaw Leszczynski. His daughter Maria became queen of France and mother of king Louis XVI. Later the castle was taken over by Prussian authorities. In 1919 the Versailles Treaty gave it back to Poland. In 1970 was taken over by the Association of Polish Mechanical Engineers (S.I.M.P). In 1994 was awarded by EUROPA NOSTRA recognition of an excellent restoration work. Source: Internet.

Recently, some articles appear in the literature (Suyal et al, 2009, Ghil et al, 2011, Tarnopolski, 2015) in relation to study sunspot, extremes events and low dimension discrete conservative system respectively and how they are related to the Hurst phenomenon.

Suyal et al (2009)[6] explain the presence of multiple Hurst exponents[7] by their resemblances (similarities, similitude) to the deterministic chaotic attractors having one or more (multiple) center of rotation in phase space. They conclude that estimating a single Hurst exponent form the data where different linear regimes exist may be improper. It is convenient to mention that time series of precipitation under climate change may have different Hurst exponents due to the non-stationary condition of the process. The authors analyzed a deterministic dynamical system represented by the Rössler system (Rössler, 1979)[8] and also the Lorenz system. It is concluded that Hurst exponent for the Lorenz system, for the value “x” (for the intensity of convection), reach value of 0.93 (first part of the linear scaling) and 0.64 for the second part of the linear scaling. The first one corresponds to the trajectory rotating about individuals fixed points while the second one is for all three fixed points.

Ghill et al (2011)[9] did a review work on extreme events, their causes and consequences. The review covers theoretical aspect of time series analysis and extreme value theory, as well as deterministic modeling of extreme events, via continuous and discrete dynamics models. Extreme events as defined by Kantz et al (2005)[10]: (1) they are rare, (2) they occur irregularly, (3) they exhibit an observable that take on an extreme value, (4) they are inherently to the system under study, rather than being due to external shocks.

The linear part of a temporal structure in the noisy part of a time series is given by the auto-correlation function (AFC); this structure is also freely referred to a persistence or memory. The so called long memory is suspected to occur in many natural processes such as river runoff (Kallache et al, 2005[11], Muldesse et al, 2007[12]) which is characterized by slow decay of the ACF. This slow decay is often referred as the Hurst phenomenon (Hurst, 1951[13]). One of the earliest schemes for estimating long range dependence (LRD) parameter is the rescaled-range statistic (R/S). It was originally proposed by Hurst (1951) when studying the Nile River low minima (Hurst 1952[14], Kondrashov et al, 2005[15]). The ACF describes the linear interdependence of two instance of a time series separated by the lag τ. A diverging sum (or infinite memory) given by

∑ ρ (τk) = ∞

It characterizes a long range dependent (LRD) process, also called long memory process or long range correlated. The example for an ACF with such a diverging sum shows an algebraic decay for a large time lags τ

Now, the spectral density [S(f)] and the ACF ρ (τ) are Fourier transform of each other. Then S(f) can also be used to describe long term memory. In this representation the LRD property reveals itself as a spectral density that diverges at zero frequency. Then, S(f) ~ |f|-βfor |f|—–> 0, withan exponent0<β = 1-ϒ<1that is the conjugate of the behavior of ρ (τ) at infinity (Rudin, 1987)[17].

The parameters β and ϒ are only two of several parameters to quantify LRD properties. A uncomplicated way is to estimate the auto correlation function from the time series and fit a power law according to ρ (τ) = τ-ϒ for τ —-> ∞ (to the largest time lags). This is done very often by least-square fitting a straight line to the autocorrelation function values plotted in a log-log coordinates. Then, the exponent found is an estimate of ϒ.

One of the initial methods to estimate long range dependence parameters is the rescaled range statistic (R/S). It was originally proposed by Hurst, 1951 when studying the Nile River flow minimum water level. The estimate parameter is now called the Hurst exponent. The R/S technique and estimation of the Hurst exponent have been extensively discussed in the long range dependence literature (Mandelbrot and Wallis, 1969[18], Salas et al, 1979[19], Salas, 1993[20], Mesa and Poveda, 1993[21], Katsev and L’Heureux, 2003[22], Koutsoyianis, 2003[23], Sakalauskiené, 2003[24], Baranik and Kristoufek, 2012[25], Fernandez-Martinez et al, 2014[26]).

In particular, Klemes (1974)[27] established in the abstract of his paper “the Hurst phenomenon is not necessarily [only] an indicator of infinite memory of a process. It can also be caused by non-stationarity in the mean and by random walks with one absorbing barrier, which often arise in natural storage systems. Attention is drawn to the fact that inferences about physical features of a process, based on operational models, can be not only inaccurate but grossly misleading”. The author concluded that the Hurst phenomenon cannot be attributed to one specific physical cause. The reasons can be because the existence of infinite memory or it can be brought about specific storage systems (an issue that is more a hydrological perspective rather that a mathematical one) or it can be a result of non-stationarity, in the process central tendency. This last attribution has been analyzed by Boes and Salas, 1978[28] in terms of the non-stationarity of the mean and the Hurst phenomenon. Four models were selected for comparison via simulation. The four models are (1) ARMA (1, 1) with standard normally distributed noise, (2) a Klemes-Potter model[29] with geometrical distributed time spans of common means, (3) Klemes-Potter model with geometrical distributed time spans of common means, exponentially distributed means with parameter λ, and standard normally distributed noise and (4) a simple case of a mixture model with geometrical distributed time span and a Bernoulli-distributed mixing distribution. The rescaled range (R/S) of partial sums was used to compare the long-term persistence of the four models. The results suggest that it is the autocorrelation function that influences the behavior of the expected rescaled range. As the authors mentioned “the similarity was expected since the rescaled range is a function of partial sums, and the central limit theorem indicates that such sums are approximately normally distributed and normal distribution are determined by their autocorrelation structure”.

As an illustration a pox diagram which is self-explanatory is presented in the following figure.

Another method is the detrended fluctuation analysis (Kantelhardt et al, 2001[30], Király and Janosi, 2004[31] ). As well as the rescaled range (R/S), it produces a heuristic estimator; hence confident intervals are not readily available, thus making statistical inference rather difficult.

Conclusions

Perhaps, the following question needs a reaction (response): What are the mechanisms that could generate persistencies and the associated Hurst phenomena? [32]

Note: Perhaps, this will be the question at the beginning and later to mention that the topic is open like it was mention at first lines.

Perhaps, one has to be clear that there are common classes of methods for identifying “persistence” characteristics in time series, those which measure autocorrelation and those which perform a Fourier, wavelet or other transformation. One common example of the former approach is the Hurst exponent (Hurst 1951). This technique breaks a time series into a number of subseries samples of increasing length and calculates the rescaled range (R/S)[33] of each subseries. The Hurst exponent is then given by the slope of the logarithm of R/S over the logarithm of the number of element in each subseries. It is based on the idea that S increases with increasing time series length, at a greater rate if the time series contain persistence. It has been an issue that long time series are required to produce a valid Hurst exponent (Wallis and O’ Connell 1973, page 363, Vogel et al, 1998). However, values obtained for many authors (e.g, Mesa, Gupta and O’Connell, 2012)) have indicated Hurst exponent values greater than 0.5. An analysis of the Hurst exponent does seem suitable from data provided from GCMs. Rocheta et al, 2014 showed a comparison of the Hurst exponent with a modified aggregated persistence score (APS) for observed record and showed that there were reasonable correlation between this two metrics. They also used to compare the Hurst exponent calculated from GCMs data with the APS. The results show that there is more scatter between the two metrics when they both identify an undersimulation of persistence in the GCMs.

Deterministic dynamical system (The Lorenz model) and Hurst Phenomenon

In 1963 Edward Lorenz was working on models to describe the convection in the earth’s atmosphere. The basic laws that govern this event are formulated as non linear differential equations which contain derivatives with respect to both space and time. Lorenz expanded the equations into a set of trigonometric functions in space and truncated the resulting infinitely many ordinary differential equation for the amplitudes of these modes to a three-dimension system. Perhaps, this is the first set of nonlinear equations known to exhibit chaotic behavior.

This now notorious system which derived as a simplified model of Rayleigh Bénard convection in the atmospheres is known as Lorenz equations and is given by the following equations

dx/dt = Pr (y-x),

dy/dt = Ra x – y – xz with Pr, Ra, b ≥ 0, (1.1)

dz/dt = xy – bz.

Where Pr is the Prandtl number, the ability to conduct heat, the parameter b is related to the aspect ratio, the ratio between the vertical and horizontal dimension of the layer describing the geometrical properties of the system. The Rayleigh number (Ra) is the varied control parameter which depend basically or most importantly, the vertical temperature gradient in the fluid or gas layer.

Figure 3: A physical interpretation of the Lorenz system can be elaborated as follow, x, y, z represent: (x) the intensity of convection motion, (y) is proportional to the temperature difference between the ascending and descending currents (same sign of x and y mean that warmer fluid is raising) and (z) is proportional to the distortion of the average vertical profile of temperature from linearity, positive values of z means strong gradient near boundaries. Two cases are represented for (x, z) Ra= 28 (above) and for (x, z) Ra=56 (below). Source: Internet Presentation.

One fixed point of the Lorenz model (1.1) is obviously the origin X1 = (0, 0, 0). In addition, we have

The Hurst phenomenon is a measure of autocorrelation (persistence and long memory).

Estimation of the Hurst exponent (H) has been determined for the “x” value with Pr=10, b=8/3 and Ra=24.74. The results indicated average Hurst exponent greater (H) than 0.5 (Mata 1991) and with Pr=16, b=4 and Ra=50 (Suyal et al 2009) which indicated on both cases a long-term dependence.

Turbulence Flow and Hurst exponent

For turbulence flow Helland and Van Atta, 1978[34], estimation of the statistical property called the ‘rescaled range’ (R/S) in grid-generated turbulence has been determined and exhibit a Hurst coefficient H = 0·5 for 43 < UT/M < 1850, where M is the grid size M and U is the mean velocity. As known, theoretically H = 0·5 for independence of two observations separated by a time interval T, and the deviation from H = 0·5 is referred to as the ‘Hurst phenomenon’. The rescaled range (R/S) obtained for grid turbulence contains an initial region UT/M < 43 of large H, approaching 1·0, corresponding approximately to the usual region of a finite non-zero autocorrelation of turbulent velocity fluctuations. For UT/M > 1850 the rescaled range breaks from H = 0·5 and rises at a significantly faster rate, H = 0·7-0·8, implying a long-term dependence or possibly non-stationarity at long times. The measured autocorrelations remain indistinguishable from zero for UT/M > 20. The break in the trend H = 0·5 is probably caused by motions on scales comparable to characteristic time scales of the wind-tunnel circulation. Rescaled-range analysis is a powerful statistical tool for determining the time scale separating the grid turbulence from the background wind-tunnel motions.

Also turbulence flow, longitudinal velocity fluctuations, for stratified flow generated in a wind tunnel has been analyzed by Salas and Mata, 1980[35] in terms of the Hurst phenomenon. An ARIMA process was found to fit the shape of the autocorrelation function.

General Circulation Models and Hurst Phenomenon

How adequately general circulation model reflect the low frequency behavior of precipitation? This issue is extremely important. It is a proper question and therefore need an appropriate response. Droughts (Pellertier and Turcotte, 1997)[36] effectively are manifestation of low frequency climate behavior. Therefore, general circulation models should be able to replicate droughts with severity and lengths consistent with historic observations and also should have the ability to predict future drought statistic (Lettenmaier and Wood, 2009)[37]; also the reliability of water supply reservoirs is closely related to persistence of below average inflow (Hurst et al, 1965[38], Douglas et al, 2002[39], Whiting, 2006[40])

Jain and Eischeid, 2008[41] and Fraedrich et al, 2009[42] report that the Hurst exponent (H) in a 1000 year simulation of the present day climate with a coupled general circulation model in a present day constant greenhouse environment agree closely with H estimated from the observations. However a paper by Rutten et al, 2009[43] analyzed global gridded observed precipitation observations (records) from the 20th century, and 20th century runs from four global circulation models and for different climate scenarios[44]. The results show that the observations had Hurst exponent values that are (considerably) larger than the values computed for precipitation from global circulation models (GCMs). This implies that climate model simulation may not adequately reflect the low frequency behavior of precipitation observed in historical records. Koutsoyiannis et al, 2008[45] concluded that GCMS do not reproduce natural inter-annual fluctuations and generally underestimated the variance and the Hurst exponent of the observed time series.

Rocheta et al, 2014[46], indicate that general circulation models (GCMs) provide reliable simulations of global and continental scale atmospheric variables, nonetheless have limited ability in simulating variables important for water resource management at regional to catchment scales. An important GCM unfairness (bias) in managing water resources infrastructure is the underrepresentation of low-frequency variability a typical characteristic to the simulation of droughts and floods. Aggregate persistence score (APS) is used to indentifies regions where GCMs weakly represent the amount of variability seen in the observed precipitation. The results show that there were (1) large spatial variations in the ability (skill)of GCMs to capture observed precipitation, (2) extensive (widespread) under simulation of rainfall persistence characteristics in GCMs, and (3) substantial improvement in rainfall persistence after applying bias correction.

As an illustration the results for Global Hurst exponent from observed precipitation is presented in figure 4, and